Prof.  J.  C.  Maxwell  on  Electric  Induct  ion,  533 
opposite  direction  to  its  motion,  and  equal  to 
m2      VRa  +  tr  +  R-t; 
AzzV  (VR^H^+R)2' 
Besides  this  retarding  force,  it  is  acted  on  by  a  force  repelling  it 
from  the  sheet,  equal  to 
4z?  R2  +  v2  +  rVR2-{-i>2 
14.  If  the  pole  moves  uniformly  in  a  circle,  the  trail  is  in  the  form 
of  a  helix,  and  the  calculation  of  its  effect  is  more  difficult ;  it  is 
easy,  however,  to  see  that,  besides  the  retarding  force  and  the  repel- 
ling force,  there  is  also  a  force  towards  the  centre  of  the  circle. 
15.  It  is  shown,  in  my  treatise  on  Electricity  and  Magnetism 
(vol.  ii.  art.  600),  that  the  currents  in  any  system  are  the  same, 
whether  the  conducting  system  or  the  inducing  system  be  in  motion, 
provided  the  relative  motion  is  the  same.  Hence  the  results  already 
given  are  directly  applicable  to  the  case  of  Arago's  rotating  disk, 
provided  the  induced  currents  are  not  sensibly  affected  by  the  limi- 
tation arising  from  the  edge  of  the  disk.  These  will  introduce  other 
sets  of  images,  which  we  shall  not  now  investigate. 
16.  The  greater  the  resistance  of  the  sheet,  whether  from  its 
thinness  or  from  the  low  conducting-power  of  its  material,  the  greater 
is  the  velocity  R.  Hence  in  most  actual  cases  R  is  very  great  com- 
pared with  vy  the  velocity  of  the  external  system,  and  the  trail  of 
images  is  nearly  normal  to  the  sheet,  and  the  induced  currents  differ 
little  from  those  which  arise  from  the  direct  action  of  the  external 
system  (see  §  1). 
1 7.  If  the  conductivity  of  the  sheet  were  infinite,  or  its  resistance 
zero,  R  would  be  zero.  The  images,  once  formed,  would  remain  sta- 
tionary, and  all  except  the  last  formed  positive  image  would  be  neu- 
tralized. Hence  the  trail  would  be  reduced  to  a  single  positive  image, 
and  the  sheet  would  exert  a  repulsive  force  —  on  the  pole,  whether 
the  pole  were  in  motion  or  at  rest. 
I  need  not  say  that  this  case  does  not  occur  in  nature  as  we  know 
it.  Something  of  the  kind  is  supposed  to  exist  in  the  interior  of  mo- 
lecules in  Weber's  Theory  of  Diamagnetism. 
Mathematical  Investigation. 
18.  Let  the  conducting  sheet  coincide  with  the  plane  of  ay,  and 
let  its  thickness  be  so  small  that  we  may  neglect  the  variation  of 
magnetic  force  at  different  points  of  the  same  normal  within  its  sub- 
stance, and  that,  for  the  same  reason,  the  only  currents  which  can 
produce  sensible  effects  are  those  which  are  parallel  to  the  surface  of 
the  sheet. 
Current-function. 
19.  We  shall  define  the  currents  in  the  sheet  by  means  of  the 
current-function  </>.     This  function  expresses  the  quantity  of  electri- 
